TY - JOUR
AB - Empirical essays of fitness landscapes suggest that they may be rugged, that is having multiple fitness peaks. Such fitness landscapes, those that have multiple peaks, necessarily have special local structures, called reciprocal sign epistasis (Poelwijk et al. in J Theor Biol 272:141–144, 2011). Here, we investigate the quantitative relationship between the number of fitness peaks and the number of reciprocal sign epistatic interactions. Previously, it has been shown (Poelwijk et al. in J Theor Biol 272:141–144, 2011) that pairwise reciprocal sign epistasis is a necessary but not sufficient condition for the existence of multiple peaks. Applying discrete Morse theory, which to our knowledge has never been used in this context, we extend this result by giving the minimal number of reciprocal sign epistatic interactions required to create a given number of peaks.
AU - Saona Urmeneta, Raimundo J
AU - Kondrashov, Fyodor
AU - Khudiakova, Kseniia
ID - 11447
IS - 8
JF - Bulletin of Mathematical Biology
KW - Computational Theory and Mathematics
KW - General Agricultural and Biological Sciences
KW - Pharmacology
KW - General Environmental Science
KW - General Biochemistry
KW - Genetics and Molecular Biology
KW - General Mathematics
KW - Immunology
KW - General Neuroscience
SN - 0092-8240
TI - Relation between the number of peaks and the number of reciprocal sign epistatic interactions
VL - 84
ER -
TY - JOUR
AB - We determine the unique factorization of some polynomials over a finite local commutative ring with identity explicitly. This solves and generalizes the main conjecture of Qian, Shi and Solé in [13]. We also give some applications to enumeration of certain generalized double circulant self-dual and linear complementary dual (LCD) codes over some finite rings together with an application in asymptotic coding theory.
AU - Köse, Seyda
AU - Özbudak, Ferruh
ID - 10842
IS - 4
JF - Cryptography and Communications
KW - Applied Mathematics
KW - Computational Theory and Mathematics
KW - Computer Networks and Communications
SN - 1936-2447
TI - Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes
VL - 14
ER -
TY - JOUR
AB - We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.
AU - Boissonnat, Jean-Daniel
AU - Kachanovich, Siargey
AU - Wintraecken, Mathijs
ID - 8940
IS - 1
JF - Discrete & Computational Geometry
KW - Theoretical Computer Science
KW - Computational Theory and Mathematics
KW - Geometry and Topology
KW - Discrete Mathematics and Combinatorics
SN - 0179-5376
TI - Triangulating submanifolds: An elementary and quantified version of Whitney’s method
VL - 66
ER -
TY - JOUR
AB - We study the problem of recovering an unknown signal 𝑥𝑥 given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator 𝑥𝑥^L and a spectral estimator 𝑥𝑥^s. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine 𝑥𝑥^L and 𝑥𝑥^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of 𝑥𝑥^L and 𝑥𝑥^s, given the limiting distribution of the signal 𝑥𝑥. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form 𝜃𝑥𝑥^L+𝑥𝑥^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s), we design and analyze an approximate message passing algorithm whose iterates give 𝑥𝑥^L and approach 𝑥𝑥^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
AU - Mondelli, Marco
AU - Thrampoulidis, Christos
AU - Venkataramanan, Ramji
ID - 10211
JF - Foundations of Computational Mathematics
KW - Applied Mathematics
KW - Computational Theory and Mathematics
KW - Computational Mathematics
KW - Analysis
SN - 1615-3375
TI - Optimal combination of linear and spectral estimators for generalized linear models
ER -
TY - JOUR
AB - Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group Sn, there exists an Sn-equivariant map X→Rn whose image avoids the diagonal {(x,x,…,x)∈Rn∣x∈R}. Previously, the special cases of this statement for certain X were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of Sn-equivariant maps from the boundary ∂Δn−1 of (n−1)-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.
AU - Avvakumov, Sergey
AU - Kudrya, Sergey
ID - 11446
IS - 3
JF - Discrete & Computational Geometry
KW - Computational Theory and Mathematics
KW - Discrete Mathematics and Combinatorics
KW - Geometry and Topology
KW - Theoretical Computer Science
SN - 0179-5376
TI - Vanishing of all equivariant obstructions and the mapping degree
VL - 66
ER -
TY - JOUR
AB - Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where “properly colored” means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.
AU - Kaveh, Kamran
AU - McAvoy, Alex
AU - Chatterjee, Krishnendu
AU - Nowak, Martin A.
ID - 8767
IS - 11
JF - PLOS Computational Biology
KW - Ecology
KW - Modelling and Simulation
KW - Computational Theory and Mathematics
KW - Genetics
KW - Ecology
KW - Evolution
KW - Behavior and Systematics
KW - Molecular Biology
KW - Cellular and Molecular Neuroscience
SN - 1553-734X
TI - The Moran process on 2-chromatic graphs
VL - 16
ER -
TY - JOUR
AB - Nuclear magnetic resonance (NMR) is a powerful tool for observing the motion of biomolecules at the atomic level. One technique, the analysis of relaxation dispersion phenomenon, is highly suited for studying the kinetics and thermodynamics of biological processes. Built on top of the relax computational environment for NMR dynamics is a new dispersion analysis designed to be comprehensive, accurate and easy-to-use. The software supports more models, both numeric and analytic, than current solutions. An automated protocol, available for scripting and driving the graphical user interface (GUI), is designed to simplify the analysis of dispersion data for NMR spectroscopists. Decreases in optimization time are granted by parallelization for running on computer clusters and by skipping an initial grid search by using parameters from one solution as the starting point for another —using analytic model results for the numeric models, taking advantage of model nesting, and using averaged non-clustered results for the clustered analysis.
AU - Morin, Sébastien
AU - Linnet, Troels E
AU - Lescanne, Mathilde
AU - Schanda, Paul
AU - Thompson, Gary S
AU - Tollinger, Martin
AU - Teilum, Kaare
AU - Gagné, Stéphane
AU - Marion, Dominique
AU - Griesinger, Christian
AU - Blackledge, Martin
AU - d’Auvergne, Edward J
ID - 8459
IS - 15
JF - Bioinformatics
KW - Statistics and Probability
KW - Computational Theory and Mathematics
KW - Biochemistry
KW - Molecular Biology
KW - Computational Mathematics
KW - Computer Science Applications
SN - 1367-4803
TI - Relax: The analysis of biomolecular kinetics and thermodynamics using NMR relaxation dispersion data
VL - 30
ER -